Structure and Zero Asymptotics of Differential Operators Associated with ${\Xi}_n$ and ${\Lambda}_n$

We study the second-order differential operators $\mathcal D_{\Xi}$ and $\mathcal D_{\Lambda}$ associated with the rescaled polynomial families $(\widetilde{\Xi}_n)$ and $(\widetilde{\Lambda}_n)$, and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum $cx-d$. We establish structural properties of $\mathcal D_{\Xi}$ and $\mathcal D_{\Lambda}$, including factorizations into first-order operators, weighted divergence forms, formal self-adjointness, and hypergeometric descriptions of the corresponding formal eigenvalue equations. We also show that both operators preserve hyperbolicity, preserve zeros in $(0,b)$ for $b\ge 1$, and preserve proper position. For the iterated polynomial sequences, we derive explicit closed formulae in terms of the auxiliary families $(\widetilde{\Xi}_n)$ and $(\widetilde{\Lambda}_n)$, prove strict interlacing of consecutive zeros under explicit conditions on $d/c$, and obtain asymptotic formulae for the normalized logarithmic derivatives. As a consequence, the associated zero counting measures converge weakly to the same limiting probability measure as in the auxiliary case.